<p>Find the tangent to the parabola \(y^2 = 4x\) at the point where it intersects the circle \(x^2 + y^2 = 5\) in the first quadrant.</p>
Step-by-Step Solution
Key Concept: Find the intersection point of the parabola and circle in the first quadrant, then use the standard tangent formula for parabola y² = 4ax: the tangent at point (at², 2at) is ty = x + at².
<p><strong>Step 1:</strong> Find intersection point. Substitute y² = 4x into x² + y² = 5:</p><p>x² + 4x = 5 ⟹ x² + 4x - 5 = 0 ⟹ (x + 5)(x - 1) = 0</p><p>So x = 1 (first quadrant), giving y² = 4, thus y = 2 (first quadrant).</p><p>Intersection point: (1, 2)</p><p><strong>Step 2:</strong> Identify parameter t for point (1, 2) on parabola y² = 4x.</p><p>For parabola y² = 4x (where 4a = 4, so a = 1), parametric form is (t², 2t).</p><p>At (1, 2): t² = 1 and 2t = 2 ⟹ t = 1</p><p><strong>Step 3:</strong> Apply tangent formula ty = x + at² with a = 1 and t = 1:</p><p>1·y = x + 1(1)² ⟹ y = x + 1 ⟹ x - y + 1 = 0</p><p>∴ Answer: C</p>
Correct Answer: C